C₂-cofiniteness of the vertex algebra V_L^+ when L is a non-degenerate even lattice

It was shown by Abe, Buhl and Dong that the vertex algebra $V_L^+$ and its irreducible weak modules satisfy the $C_2$-cofiniteness condition when $L$ is a positive definite even lattice. In this paper, we extend their results by showing that the vertex algebra $V_L^+$ and its irreducible weak modules are $C_2$-cofinite when $L$ is a negative definite even lattice and when $L$ is a non-degenerate even lattice that is neither negative definite nor positive definite.